scholarly journals The Transition to Equilibrium in a System with Gravitationally Interacting Particles. I. Temperature Relaxation

2021 ◽  
pp. 21-47
Author(s):  
A. M. Boichenko ◽  
M. S. Klenovskii
Keyword(s):  
Distribution Function ◽  
Canonical Form ◽  
Interacting Particles ◽  
Huge Number ◽  
Equilibrium Modeling ◽  
Statistical Effect ◽  
Statistical Behavior ◽  
Temperature Relaxation ◽  
Theoretical Results ◽  
Number Of Particles

The distribution function of systems in equilibrium must have the canonical form of the Gibbs distribution. To substantiate this behavior of systems, attempts have been made for more than 100 years to involve their mechanical behavior. In other words, it seems that a huge number of particles of the medium as a result of interaction with each other according to dynamic laws, is able to explain the statistical behavior of systems during their transition to equilibrium. Modeling of gravitationally interacting particles is carried out and it is shown that in this case, the distribution function does not evolve to the canonical form. Earlier, the same results were obtained for classical Coulomb plasma. On the other hand, such a statistical effect as relaxation is well described by the dynamic behavior of the system, and the simulation data are in agreement with the known theoretical results obtained in various statistical approaches.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Particle size distribution of sawdust and wood shavings mixtures

2010 ◽  
Vol 56 (No. 4) ◽  
pp. 154-158 ◽  
Author(s):  
T. Vítěz ◽  
P. Trávníček
Keyword(s):  
Experimental Data ◽  
Particle Size ◽  
Distribution Function ◽  
Network Analysis ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Mass Fraction ◽  
Particle Sizes ◽  
Wood Shavings ◽  
Number Of Particles

Particle size distribution of the sample of waste sawdust and wood shavings mixtures were made with two commonly used methods of mathematical models by Rosin-Rammler (RR model) and by Gates-Gaudin-Schuhmann (GGS model).On the basis of network analysis distribution function F (d) (mass fraction) and density function f (d) (number of particles captured between two screens) were obtained. Experimental data were evaluated using the RR model and GGS model, both models were compared. Better results were achieved with GGS model, which leads to a more accurate separation of the different particle sizes in order to obtain a better industrial profit of the material.

A LATTICE BGK MODEL FOR SIMULATING SOLITARY WAVES OF THE COMBINED KdV–MKDV EQUATION

2011 ◽  
Vol 25 (04) ◽  
pp. 589-597 ◽  
Author(s):  
CHANGFENG MA
Keyword(s):  
Distribution Function ◽  
Solitary Waves ◽  
Equilibrium Distribution ◽  
Local Equilibrium ◽  
Mkdv Equation ◽  
Equilibrium Distribution Function ◽  
Bgk Model ◽  
Theoretical Results

A lattice BGK model for simulating solitary waves of the combined KdV–MKDV equation, ut+αuux-βu2ux+δuxxx = 0, is established. The tunable parameters in Chapman–Enskog expansion of the local equilibrium distribution function are determined by the coefficient of the combined KdV–MKDV equation. Simulating results fit close in with the theoretical results.

scholarly journals Transverse transport properties of a charged drop in an electric field

2015 ◽  
Vol 24 (05) ◽  
pp. 1550034 ◽  
Author(s):  
S. Bondarenko ◽  
K. Komoshvili
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
Transport Properties ◽  
High Energy ◽  
Time Dependent ◽  
Charged Droplet ◽  
Interacting Particles ◽  
Charged Drop ◽  
Weakly Interacting ◽  
Weakly Interacting Particles

Transport properties of a charged droplet of weakly interacting particles in transverse electric field are investigated. Nonequilibrium, time-dependent distribution function which describes a process of the droplet transverse evolution with constant entropy in the field is calculated. With the help of this distribution function, shear viscosity coefficients in the transverse plane are calculated as well. They are found to be dependent on the ratio of the potential energy of the droplet in the electric field to the kinetic energy of the droplet; for weakly interacting particles, this parameter is small. Additionally, these coefficients are time-dependent and change during the hydrodynamical state of the droplet's expansion. Applicability of the results to the description of initial states of quark–gluon plasma (QGP) obtained in high-energy interactions of nuclei is also discussed.

Stochastic processes relating to particles distributed in a continuous infinity of states

1950 ◽  
Vol 46 (4) ◽  
pp. 595-602 ◽  
Author(s):  
Alladi Ramakrishnan
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Stochastic Processes ◽  
Stochastic Variable ◽  
Stochastic Problems ◽  
Image Position ◽  
Number Of Particles

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E-space were discrete and the states represented by E1, E2, …, then it would be possible to define a functionrepresenting the probability that there are ν1 particles in E1, ν2 particles in E2, …, at t. The variation of π with t is governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variableswith the help of the π function which yields also the correlation between the stochastic variables νi.

scholarly journals Theoretical Analysis of Label Distribution Learning

2019 ◽  
Vol 33 ◽  
pp. 5256-5263 ◽  
Author(s):  
Jing Wang ◽  
Xin Geng
Keyword(s):  
Distribution Function ◽  
Conditional Probability Distribution ◽  
Probability Bounds ◽  
Real World Applications ◽  
Label Distribution Learning ◽  
Definition Of ◽  
Label Distribution ◽  
Risk Bounds ◽  
Dependent Error ◽  
Theoretical Results

As a novel learning paradigm, label distribution learning (LDL) explicitly models label ambiguity with the definition of label description degree. Although lots of work has been done to deal with real-world applications, theoretical results on LDL remain unexplored. In this paper, we rethink LDL from theoretical aspects, towards analyzing learnability of LDL. Firstly, risk bounds for three representative LDL algorithms (AA-kNN, AA-BP and SA-ME) are provided. For AA-kNN, Lipschitzness of the label distribution function is assumed to bound the risk, and for AA-BP and SA-ME, rademacher complexity is utilized to give data-dependent risk bounds. Secondly, a generalized plug-in decision theorem is proposed to understand the relation between LDL and classification, uncovering that approximation to the conditional probability distribution function in absolute loss guarantees approaching to the optimal classifier, and also data-dependent error probability bounds are presented for the corresponding LDL algorithms to perform classification. As far as we know, this is perhaps the first research on theory of LDL.

scholarly journals How many particles make up a chaotic many-body quantum system?

2021 ◽  
Vol 10 (4) ◽  
Author(s):  
Guy Zisling ◽  
Lea Santos ◽  
Yevgeny Bar Lev
Keyword(s):  
Long Range ◽  
Quantum Chaos ◽  
Short Range ◽  
Many Body ◽  
Interacting Particles ◽  
One Dimensional ◽  
Long Range Interactions ◽  
Minimum Number ◽  
Many Particles ◽  
Number Of Particles

We numerically investigate the minimum number of interacting particles, which is required for the onset of strong chaos in quantum systems on a one-dimensional lattice with short-range and long-range interactions. We consider multiple system sizes which are at least three times larger than the number of particles and find that robust signatures of quantum chaos emerge for as few as 4 particles in the case of short-range interactions and as few as 3 particles for long-range interactions, and without any apparent dependence on the size of the system.

Sign in / Sign up

Export Citation Format

Share Document